where \(x\)  is a classical input, \(\operatorname{{\mathcal {E}}}\)  is an arbitrary (possibly unknown) completely positive and trace preserving (CPTP) map, and \(O\)  is a known observable.
We emphasize that model (REF ) encompasses any physical process that takes a classical input and produces a real number as output. Therefore, this general setup also includes learning classical functions, such as functions generated by neural networks [1]}. It also encompasses many classical ML tasks, such as classification and regression [2]}.
